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Problems for 2015 AIM Workshop on Dynamical Algebraic Combinatorics

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Problems for 2015 AIM Workshop on Dynamical Algebraic Combinatorics Problems for 2015 AIM workshop on Dynamical Algebraic Combinatorics Jim Propp, Tom Roby, Jessica Striker, Nathan Williams May 29, 2015 Contents 1 Homomesies 2 1.1 Classifying homomesies . .2 1.2 Telescoping . .2 1.3 Dynamical closures . .3 2 Combinatorial Problems 4 2.1 The middle runner problem . .4 2.2 Cores . .4 2.3 Perfect matchings . .5 2.4 Resonance . .5 2.5 Undiscovered combinatorial models . .6 2.5.1 The 3n − 2 Problem . .6 2.5.2 Multi-noncrossing models . .7 2.6 Products of chains . .7 3 Coxeter-theoretic Problems 8 3.1 Bijactions in Cataland . .8 3.2 Nonnesting Cataland Lifts . .8 3.3 Coincidental Types . .9 3.4 Hurwitz Actions on Factorizations of c ............... 10 4 Piecewise-Linear and Birational Toggles 11 4.1 Order polytope promotion and rowmotion . 11 4.2 Birational rowmotion on G=P .................... 11 4.3 When is birational rowmotion periodic? . 11 4.4 Order polytopes and P -partitions . 13 4.5 Cluster algebras and birational toggling . 13 4.6 Gelfand-Tsetlin triangles . 14 4.7 The birational toggle group . 14 1 5 Generalized Toggling 15 5.1 Generalized toggling from the bottom up . 16 5.2 Generalized toggling from the top down . 16 5.3 Subset toggling . 16 5.4 Toggling noncrossing partitions . 17 Note: For discussion of antichains, order ideals, and J(P ) (the set of order ide- als of a poset P ), see Enumerative Combinatorics I by Richard Stanley. For discussion of O(P ) (the order polytope of a poset P ), see Two Poset Polytopes by Richard Stanley. For discussion of promotion and rowmotion, see Promo- tion and Rowmotion by Jessica Striker and Nathan Williams. For discussion of Panyushev complementation, see On orbits of antichains of positive roots by Dmitri Panyushev. For discussion of the concepts of homomesy and 0-mesy, see Homomesy in products of two chains by James Propp and Tom Roby. For dis- cussion of piecewise linear and birational liftings of promotion and rowmotion, see Piecewise-linear and birational toggling by David Einstein and James Propp. For discussion of ASMs (alternating sign matrices) and Wieland’s gyration ac- tion on ASMs, see The many faces of alternating-sign matrices by James Propp and Promotion and Rowmotion by Jessica Striker and Nathan Williams. For discussion of the CSP (cyclic sieving phenomenon) see What is cyclic sieving? by Victor Reiner, Dennis Stanton, and Dennis White. 1 Homomesies 1.1 Classifying homomesies Let P be a finite poset and T : J(P ) ! J(P ) some invertible action on the set of order ideals of P . Given an order ideal I 2 J(P ) and an element x 2 P , let 1x(I) be 1 or 0 according to whether or not x 2 I, and let V be the real vector space of functions from J(P ) to R spanned by the function 1x. Let V0 be the subspace of V consisting of all 0-mesies, that is, functions f : J(P ) ! R such P that I2O f(I) = 0 for every T -orbit O in J(P ). Problem 1.1. What is the dimension of V0, and what is a natural basis for it? We know the answer when P is a product of two chains and T is rowmotion or promotion, but not for rowmotion and promotion of other posets. One can ask the same question with antichains instead of order ideals. Again, we know the answer when P is a product of two chains and T is rowmotion or promotion, but not for the rowmotion and promotion actions on other posets. 1.2 Telescoping Let T be an action on a set X all of whose orbits are finite, and let f be some real-valued 0-mesy of (X; T ) (that is, a real-valued function on X whose average value on each T -orbit is 0, or equivalently, whose sum on each T -orbit is 0), The 2 most straightforward way to prove that the sum of f over every T -orbit is zero is to find a way to exhibit a function g on X for which f(x) = g(x)−g(T (x)) for all x 2 X; for in that case 0-mesy is just a matter of telescoping. And indeed, some of the known proofs of homomesy rely upon similar tricks. This prompts one to ask: To what extent can proofs of homomesy results (especially 0-mesy results) be simplified by finding constructions of such “integrals” g? Note that for any 0-mesic function f there are typically many functions g satisfying the relation f(x) = g(x) − g(T (x)); the problem is the following. Problem 1.2. Find simple and uniform constructions for such functions g for a wide range of 0-mesies f. 1.3 Dynamical closures In most cases of combinatorial interest, the “feature space” V is not preserved by the map T ; that is, for f : X ! R a function in V , the time-shifted function f ◦T : X ! R typically is not in V . It therefore seems natural, when T is of finite order (order n, say), to replace V by a larger but still finite-dimensional space V T (the “dynamical closure” of V ) defined as the smallest set of functions that contains V and contains f ◦ T whenever it contains f. Indeed, many important dynamical properties of (X; T; V ) reduce to linear (or more generally affine) relations satisfied in V T : invariance asserts that for some particular f 2 V , f equals f ◦ T ; homomesy asserts that for some particular f 2 V , f + f ◦ T + f ◦ T 2 + ··· + f ◦ T n−1 equals a constant function; and reciprocity asserts that for some particular f; g 2 V and some particular k, f + g ◦ T k is the zero function. Also (to give an example that seems to be very narrow but whose seeming narrowness may be merely a reflection of our ignorance of analogous behavior in other dynamical systems), we may note that the combinatorial fact that underlies the existence of the Armstrong-Stanley way of looking at Panyushev complementation of antichains in [a] × [b], namely, the fact that A contains an element in the ith fiber if and only if the Panyushev complement of A contains an element in the (i + 1)st fiber (as long as i < a), can also be expressed as a linear relation in V T . T Here is an outline for how to approach V systematically. Letting f1; : : : ; fN T T denote a basis for V , define V (X) to be the set of N-tuples (f1(x); : : : ; fN (x)) as x varies over X. Figuring out what linear relations are satisfied by the T functions f1; : : : ; fN when they are restricted to V (X) is equivalent to: Problem 1.3. Characterize the affine closure of V T (X) in V T . One could explore this computationally for specific dynamical systems (X; T ), and infer patterns that would lead to conjectures. The cases to start with are rowmotion and promotion on [a] × [b], since a wealth of homomesy and reci- procity relations are known. Are there invariance relations as well, or other sorts of affine relations satisfied by V T (X) that don’t follow from known homo- mesy and reciprocity relations? 3 2 Combinatorial Problems 2.1 The middle runner problem Imagine n runners on a circular track, moving at various non-zero speeds (for simplicity, assume that the speeds are commensurable). The track has a starting line for counting laps (though runners are not required to start at the starting line), and the position of each runner at every instant is written as a number between 0 and 1 (representing how much of his/her current lap the runner has completed). At any instant t, let p1(t) ≤ p2(t) ≤ · · · ≤ pn(t) be the sorted positions of the runners, and for 1 ≤ i ≤ n let pi denote the average value of pi(t) over the course of one full period (the time it takes for all the runners to return to where they respectively started, which is just the lcm of the periods of all the runners). Conjecture 2.1. For i + j = n + 1, pi + pj = 1. The middle runner problem takes its name from the special case in which n is odd and i = j = (n + 1)=2: it says that the position of the middle runner is 1/2 on average. The case in which all runners have the same speed has been solved, but the general case remains open. 2.2 Cores There are bijection between simultaneous (a; b)-cores, lattice paths in the tri- angular region with vertices (0; 0); (a; 0); (a; b), and certain (a; b)-noncrossing 1 a+b partitions [AHJ14, ARW13]. There are a+b b such objects. Armstrong conjectured that the average size jcj of an (a; b)-core c was (a − 1)(b − 1)(a + b − 1) : 24 This was proved in the Catalan case in [SZ13], the Fuss-Catalan case in [Agg14], and (very recently!) in full generality (using the polynomial method and Er- hart theory—cores are naturally points in the root lattice in a dilation of the fundamental alcove in affine type A) in [Joh15]. Thus, the number and av- erage size correspond to the 0th and 1st moments of the generating function P jcj c an (a;b)−core q : Problem 2.2. Find and prove formulas for higher moments of cores: X jcji: c an (a;b)−core One can rotate an (a; b)-noncrossing partition, which can be modeled using toggles on rational slope lattice paths.
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